A094649 -id:A094649 - cp's OEIS frontend

A094648 An accelerator sequence for Catalan's constant.

3, -1, 5, -4, 13, -16, 38, -57, 117, -193, 370, -639, 1186, -2094, 3827, -6829, 12389, -22220, 40169, -72220, 130338, -234609, 423065, -761945, 1373466, -2474291, 4459278, -8034394, 14478659, -26088169, 47011093, -84708772, 152642789, -275049240

Offset: 0

Paul Barry, May 18 2004

The pair A094648 and the alternating sequence A033304 when joined form a two-sided sequence defined by the recurrence formula x(n+3) + x(n+2) - 2x(n+1) - x(n) = 0, n in Z, x(-1)=-2, x(0)=3, x(1)=-1 - for details see Witula's comments to A033304. - Roman Witula, Jul 25 2012
From Roman Witula, Aug 09 2012: (Start)
There exist two interesting subsequences b(n) and c(n) of the given above sequence x(n) defined by the following relations: b(n)=a(2^n) and c(n)=x(-2^n). These subsequences satisfy the following system of recurrence equations:
b(n+1)=b(n)^2-2*c(n), and c(n+1)=c(n)^2-2*b(n),
which easily follow from the general identity: x(n)^2=x(2*n)-2*x(-n), n in Z. We note that b(0)=-1, b(1)=5, b(2)=13, b(3)=117, c(0)=-2, c(1)=6, c(2)=26, c(3)=650. From the above system we deduce that all b(n) are odd, whereas all c(n) are even. Moreover we obtain c(n+1)-b(n+1)=(c(n)-b(n))*(b(n)+c(n)+2), which yields b(n+1)-c(n+1)=product{k=1,..,n}(b(k)+c(k)+2)=13*product{k=2,..,n}(b(k)+c(k)+2)=13^2*41*product{k=3,..,n}(b(k)+c(k)+2). It follows that b(n)-c(n) is divisible by 13^2*41 for every n=3,4,..., and after using the above system again each b(n) and c(n), for n=2,3,..., is divisible by 13. (End)
If we set W(n):=3*A077998(n)-A006054(n+1)-A006054(n), n=0,1,..., then a(n)=(W(n)^2-W(2*n))/2 and W(n) = (-c(1))^(-n) + (-c(2))^(-n) + (-c(4))^(-n) = (-c(1)*c(2))^n + (-c(1)*c(4))^n + (-c(2)*c(4))^n = (-1-c(1))^n + (-1-c(2))^n + (-1-c(4))^n, where c(j):=2*cos(2*Pi*j/7) - for the proof see Witula-Slota-Warzynski's paper. Moreover it follows from the comment at the top and from comments to A033304 that W(n+1)=A033304(n)=(-1)^(n+1)*x(-n-1). - Roman Witula, Aug 11 2012
The following trigonometric type identitities hold true: (1) -a(n-1)-a(n) = c(1)*c(2)^n + c(2)*c(4)^n + c(4)*c(1)^n and (2) a(n)-a(n+2) = c(4)*c(2)^(n+1) + c(1)*c(4)^(n+1) + c(2)*c(1)^(n+1), where a(-1)=-2 and c(j) is defined as above (see also the respective comment to A033304). For the proof see Remark 6 in Witula's paper. - Roman Witula, Aug 14 2012
It can be proved that A033304(n-1)*(-1)^n = (a(n)^2 - a(2*n))/2, n=1,2,... - Roman Witula, Sep 30 2012
With respect to the form of the trigonometric formulas describing a(n), we call this sequence the Berndt-type sequence number 19 for the argument 2*Pi/7. The A-numbers of other Berndt-type sequences numbers are given in below. - Roman Witula, Sep 30 2012

			We have a(17) = a(19) + 50000, a(4) + a(5) = -3, 2*a(7) + a(8) = 3, and 2*a(9) + a(10) = a(5). - _Roman Witula_, Sep 14 2012

G. C. Greubel, Table of n, a(n) for n = 0..3900
A. Akbary and Q. Wang, On some permutation polynomials over finite fields, International Journal of Mathematics and Mathematical Sciences, 2005:16 (2005) 2631-2640.
A. Akbary and Q. Wang, A generalized Lucas sequence and permutation binomials, Proceeding of the American Mathematical Society, 134 (1) (2006), 15-22.
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, The Ramanujan Journal, Vol. 3, Issue 2, 1999, pp. 159-173
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, arXiv:0706.0356 [math.CA], 2007.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (-1,2,1).

Cf. A000032, A094649, A094650.

Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094429, A094430, A217274.

I:=[3,-1,5]; [n le 3 select I[n]  else -Self(n-1)+2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 25 2015

CoefficientList[ Series[(3 + 2x - 2x^2)/(1 + x - 2x^2 - x^3), {x, 0, 33}], x] (* Robert G. Wilson v, May 24 2004 *)
a[n_] := Round[(2Sin[3Pi/14])^n + (-2Sin[Pi/14])^n + (-2Cos[Pi/7])^n]; Table[ a[n], {n, 0, 33}] (* Robert G. Wilson v, May 24 2004 *)
LinearRecurrence[{-1,2,1}, {3,-1,5}, 50] (* Roman Witula, Aug 09 2012 *)

x='x+O('x^30); Vec((3+2*x-2*x^2)/(1+x-2*x^2-x^3)) \\ G. C. Greubel, May 09 2018

G.f.: (3+2*x-2*x^2)/(1+x-2*x^2-x^3);
a(n) = (2*sin(3*Pi/14))^n+(-2*sin(Pi/14))^n+(-2*cos(Pi/7))^n.
a(p) == -1 mod(p), p prime. - Philippe Deléham, Oct 03 2009
a(n) = (2*cos(2*Pi/7))^n + (2*cos(4*Pi/7))^n + (2*cos(8*Pi/7))^n, which is equivalent to the formula given above (for analogous sums with sines see A215493 and A215494). Moreover we have a(n+3) + a(n+2) - 2a(n+1) - a(n) = 0 - for the proof see Witula-Slota's paper. - Roman Witula, Jul 24 2012
a(n) = 3*(-1)^n*A006053(n+2) +2*A078038(n-1). - R. J. Mathar, Nov 03 2020

A216605 Expansion of g.f. (6 + 5x - 20x^2 - 12x^3 + 12x^4 + 3x^5)/(1 + x - 5x^2 - 4x^3 + 6x^4 + 3*x^5 - x^6).

Original entry on oeis.org

6, -1, 11, -4, 31, -16, 98, -64, 327, -256, 1126, -1024, 3958, -4083, 14116, -16189, 50887, -63768, 184958, -249547, 676626, -970771, 2488156, -3756867, 9188406, -14474916, 34049481, -55564474, 126540536, -212637571, 471398623, -811660849, 1759603367

Offset: 0

Roman Witula, Sep 10 2012

Previous name was: a(n) is equal to the rational part (considering of the ring Z(sqrt(13))) of the numbers A(n) = ((sqrt(13) - 1)/2)*A(n-1) + A(n-2) + ((3-sqrt(13))/2)*A(n-3), with A(0) = 6, A(1) = sqrt(13) - 1, and A(2) = 11 - sqrt(13).
The Berndt-type sequence number 1 for the argument 2*Pi/13 defined by the following relation: a(n) + b(n)*sqrt(13) = A(n) = 2*(c(1)^n + c(3)^n + c(4)^n), where c(j) := 2*cos(2*Pi*j/13), j=1..6. The b(n) = A216486(n), n=0,1,..., are all positive integers. We note that we also have a(n) - b(n)*sqrt(13) = B(n) = 2*(c(2)^n + c(5)^n + c(6)^n) and the following recurrence relation holds: B(n) = -((sqrt(13)+ 1)/2)*B(n-1) + B(n-2) + ((3+sqrt(13))/2)*B(n-3), with B(0) = 6, B(1) = -sqrt(13) - 1, and B(2) = 11 + sqrt(13).
We note that 4*a(n) - a(n+2) is divisible by 13 for every n = 0,1,... .

			We have 4*a(2*n-1)=a(2*n+1) for every n = 1,2,...,5 and a(13) - 4*a(11) = 13. Further we have c(1)^3 + c(3)^3 + c(4)^3 = 4*(c(1) + c(3) + c(4)) since A(3) = 4*sqrt(13) - 4, c(2)^3 + c(5)^3 + c(6)^3 = 4*(c(2) + c(5) + c(6)) since B(3) = - 4*sqrt(13) - 4, 2 + c(1)^4 + c(3)^4 + c(4)^4 = 3*(c(1)^2 + c(3)^2 + c(4)^2) and 2 + c(2)^4 + c(5)^4 + c(6)^4 = 3*(c(2)^2 + c(5)^2 + c(6)^2).

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and Their Applications, Congressus Numerantium, 201 (2010), 89-107.
R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Akbary and Q. Wang, A generalized Lucas sequence and permutation binomials, Proc. Am. Math. Soc. 134 (2006) 15-22. Sequence a(n) with l=13 is the unsigned version.
Russell A. Gordon, Lucas Type Sequences and Sums of Binomial Coefficients, Integers (2023) Vol 23, Art. No. A84. See p. 21.
R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.
Index entries for linear recurrences with constant coefficients, signature (-1,5,4,-6,-3,1).

Cf. A094649, A189234, A216486.

I:=[6,-1,11,-4,31,-16]; [n le 6 select I[n] else -Self(n-1)+5*Self(n-2)+4*Self(n-3)-6*Self(n-4)-3*Self(n-5)+Self(n-6): n in [1..35]]; // Vincenzo Librandi, Aug 30 2017

LinearRecurrence[{-1, 5, 4, -6, -3, 1}, {6, -1, 11, -4, 31, -16}, 30]
CoefficientList[Series[(6 + 5 x - 20 x^2 - 12 x^3 + 12 x^4 + 3 x^5)/(1 + x - 5 x^2 - 4 x^3 + 6 x^4 + 3 x^5 - x^6), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 30 2017 *)

G.f.: (6 + 5*x - 20*x^2 - 12*x^3 + 12*x^4 + 3*x^5)/(1 + x - 5*x^2 - 4*x^3 + 6*x^4 + 3*x^5 - x^6). - Bruno Berselli, Sep 11 2012

a(n) = -a(n-1) + 5*a(n-2) + 4*a(n-3) - 6*a(n-4) - 3*a(n-5) + a(n-6), which from the following decomposition can be generated (see Witula-Slota's and Witula's references for details): X^6 + X^5 - 5*X^4 - 4*X^3 + 6*X^2 + 3*X - 1 = ((X - c(1))*(X - c(3))*(X - c(4)))*((X - c(2))*(X - c(5))*(X - c(6))) = (X^3 + ((1 - sqrt(13))/2)*X^2 - X + (sqrt(13) - 3)/2)*(X^3 + ((1 + sqrt(13))/2)*X^2 - X - (sqrt(13) + 3)/2). - Roman Witula, Sep 11 2012

New name using existing g.f. from Joerg Arndt, Feb 15 2024

A094650 An accelerator sequence for Catalan's constant.

Original entry on oeis.org

5, -1, 9, -4, 25, -16, 78, -64, 257, -256, 874, -1013, 3034, -3953, 10684, -15229, 38017, -58056, 136338, -219508, 491870, -824737, 1782735, -3083887, 6484514, -11489516, 23652443, -42688039, 86459608, -158270401, 316576903, -585868009, 1160673633

Offset: 0

Paul Barry, May 18 2004

A. Akbary and Q. Wang, On some permutation polynomials over finite fields, International Journal of Mathematics and Mathematical Sciences, 2005:16 (2005) 2631-2640.
A. Akbary and Q. Wang, A generalized Lucas sequence and permutation binomials, Proceeding of the American Mathematical Society, 134 (1) (2006), 15-22.
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, The Ramanujan Journal, Vol. 3, Issue 2, 1999, pp. 159-173
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, arXiv:0706.0356 [math.CA], 2007.
Russell A. Gordon, Lucas Type Sequences and Sums of Binomial Coefficients, Integers (2023) Vol 23, Art. No. A84. See p. 21.
Index entries for linear recurrences with constant coefficients, signature (-1,4,3,-3,-1).

Cf. A000032, A094648, A094649.

LinearRecurrence[{-1, 4, 3, -3, -1}, {5, -1, 9, -4, 25}, 33] (* Jean-François Alcover, Sep 21 2017 *)

Vec((5+4*x-12*x^2-6*x^3+3*x^4)/(1+x-4*x^2-3*x^3+3*x^4+x^5) + O(x^40)) \\ Michel Marcus, Jul 25 2015

G.f.: (5+4x-12x^2-6x^3+3x^4)/(1+x-4x^2-3x^3+3x^4+x^5).

a(n) = (2*cos(2*Pi/11))^n + (-2*cos(Pi/11))^n + (-2*sin(5*Pi/22))^n +(2*sin(3*Pi/22))^n + (-2*sin(Pi/22))^n.

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 3, 1, 5, 1, 5, 4, 1, 6, 1, 7, 4, 7, 1, 8, 1, 11, 4, 19, 16, 18, 1, 9, 1, 13, 4, 25, 16, 38, 29, 1, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 12, 1, 19, 4, 43, 16, 98, 64, 187, 193, 123, 1

Offset: 1

L. Edson Jeffery, Jan 12 2013

Antidiagonal sums: {1,3,5,9,16,26,46,78,136,...}.

			Array begins as
.1..1...1..1...1...1
.2..1...3..4...7..11
.3..1...5..4..13..16
.4..1...7..4..19..16
.5..1...9..4..25..16
.6..1..11..4..31..16

Rows: cf. A000012, A000032, A094649, A189234, A216605, etc.

Cf. A185095, A186740.

T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.
Empirical g.f. for row n: F(x) = (Sum_{u=0..n-1} A122765(n,n-1-u)*x^u)/(Sum_{v=0..n} A108299(n,v)*x^v).
Empirical: odd column first differences tend to A000984 = {1, 2, 6, 20, 70, 252, ...} (central binomial coefficients).

A376498 Array read by ascending antidiagonals: A(n, k) = 2^kSum_{j=1..n} cos((2j - 1)Pi/(2n + 1))^k.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 1, 3, 1, 0, 5, 1, 5, 4, 1, 0, 6, 1, 7, 4, 7, 1, 0, 7, 1, 9, 4, 13, 11, 1, 0, 8, 1, 11, 4, 19, 16, 18, 1, 0, 9, 1, 13, 4, 25, 16, 38, 29, 1, 0, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 0, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 0

Offset: 0

Cheng-Jun Li, Sep 25 2024

It is only a conjecture that the A(n, k) are always integers.

			Array starts:
[0] 0, 0,  0, 0,  0,  0,   0,  0,   0,   0,    0,    0, ...  [A000004]
[1] 1, 1,  1, 1,  1,  1,   1,  1,   1,   1,    1,    1, ...  [A000012]
[2] 2, 1,  3, 4,  7, 11,  18, 29,  47,  76,  123,  199, ...  [A000032]
[3] 3, 1,  5, 4, 13, 16,  38, 57, 117, 193,  370,  639, ...  [A096975]
[4] 4, 1,  7, 4, 19, 16,  58, 64, 187, 247,  622,  925, ...  [A094649]
[5] 5, 1,  9, 4, 25, 16,  78, 64, 257, 256,  874, 1013, ...  [A189234]
[6] 6, 1, 11, 4, 31, 16,  98, 64, 327, 256, 1126, 1024, ...  [A216605]
[7] 7, 1, 13, 4, 37, 16, 118, 64, 397, 256, 1378, 1024, ...
[8] 8, 1, 15, 4, 43, 16, 138, 64, 467, 256, 1630, 1024, ...
[9] 9, 1, 17, 4, 49, 16, 158, 64, 537, 256, 1882, 1024, ...

Rows: A000004 (n=0), A000012 (n=1), A000032 (n=2), A096975 (n=3), A094649 (n=4), A189234 (n=5), A216605 (n=6, with alternate signs).
Columns: A001477 (k=0), A057427 (k=1).
Cf. A180870.

A(n, k) = 2^k*sum(j=1, n, (cos((2*j-1)*Pi/(2*n+1)))^k, x=0)

A(n + k, 2*k - 1) = A(k, 2*k-1) = 4^(k-1).

Let P_n(x) be the polynomial: Sum_{k=0..n} x^k*A180870(n, k). Let R_n(x) be the polynomial Product_{k=0..n} x-Roots(P_n, k)^m. A(n, k) = abs([x^1] R_n(x))/2^(m*(n-1)), for n > 0. - Thomas Scheuerle, Oct 07 2024

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A094648 An accelerator sequence for Catalan's constant.

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A216605 Expansion of g.f. (6 + 5*x - 20*x^2 - 12*x^3 + 12*x^4 + 3*x^5)/(1 + x - 5*x^2 - 4*x^3 + 6*x^4 + 3*x^5 - x^6).

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A094650 An accelerator sequence for Catalan's constant.

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A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.

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A376498 Array read by ascending antidiagonals: A(n, k) = 2^k*Sum_{j=1..n} cos((2*j - 1)*Pi/(2*n + 1))^k.

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A216605 Expansion of g.f. (6 + 5x - 20x^2 - 12x^3 + 12x^4 + 3x^5)/(1 + x - 5x^2 - 4x^3 + 6x^4 + 3*x^5 - x^6).

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.

A376498 Array read by ascending antidiagonals: A(n, k) = 2^kSum_{j=1..n} cos((2j - 1)Pi/(2n + 1))^k.